Kernel and eigenfunction estimates for some second order elliptic operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:387

Article

Ouhabaz, El Maati²  Rhandi, Abdelaziz¹ 

[1] Univ Salerno, Dept Math, I-84084 Fisciano, SA, Italy

[2] Univ Bordeaux 1, Inst Math IMB, Equipe Anal, F-33405 Talence, France

关键词: Heat kernels;    Schrodinger operators;    Eigenfunctions;    Log-Sobolev inequality;   

DOI  :  10.1016/j.jmaa.2011.09.045

来源: Elsevier

【 摘 要 】

For a potential V such that V(x) vertical bar x vertical bar(alpha) with alpha > 2 we prove that the heat kernel k(t)(x, y) associated to the uniformly elliptic operator A = -Sigma(n)(j,k=1) partial derivative(k)(a(jk)partial derivative(j)) + V satisfies the estimate k(t)(x, y) <= Ce-mu 0t e(ct-b) (e(-2 root theta/alpha+2 vertical bar x vertical bar 1+alpha/2)/vertical bar x vertical bar(alpha/4+n-1/2))(e(-2 root theta/alpha+2 vertical bar y vertical bar 1+alpha/2)/vertical bar y vertical bar(alpha/4+n-1/2)) for large x, y is an element of R-n and all t > 0. Here 0 < theta <= 1 is an appropriate constant, b > alpha+2/alpha-2 and mu(0) is the first eigenvalue of A. We also obtain an estimate for large vertical bar x vertical bar of the eigenfunctions of A. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

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